A few questions on abelian and normal subgroups

[blahblah8724]

Just to help my understanding...


1) Can you have an abelian subgroup of a group which isn't normal?

2) Can you have a normal subgroup which isn't abelian?

Cheers!

 

[micromass]

1) Can you list me all the subgroups of $D_6$?? (the dihedral group with 6 elements). Which one of these subgroups is abelian, which one is normal??

2) Can you give a normal subgroup of $S_n$?? (the symmetric group on n elements)

 

[mathwonk]

are cyclic subgroups abelian?

 

[lavinia]

Just to help my understanding...


1) Can you have an abelian subgroup of a group which isn't normal?

2) Can you have a normal subgroup which isn't abelian?

Cheers!

I think that for this sort of question you should try to answer it for yourself by looking at examples. Take a couple of invertible matrices of finite order and look at the groups that they generate.

 

[Deveno]

{e, (1 2)} is a non-normal subgroup of S3, which is abelian, as all groups of order 2 are.

A₄ is a non-abelian subgroup of S₄, which is normal (as any subgroup of index 2 is).

there is almost no connection between the concept of normal and abelian...with one exception.

IF G is abelian, then any subgroup is normal since gh = hg → ghg^-1 = h

and thus gHg^-1= H, for any subgroup H.

even though it is possible to define "product sets" in groups, such as:

HK = {hk : h in H, k in K}, one can't treat the "sets" as if they were "elements", in general.

for example if G = HK = KH, one cannot conclude that G is abelian.

one way of thinking about it, is that "abelian groups" are "nice", there is no need to worry about "which subgroups are normal", we just need the concept "subgroup".

but for groups in general, "normal subgroups" are special, we can factor them out.

for a dihedral group, the rotation group is normal, "factoring it out" still leaves us in the same plane we started in. the reflection subgroups are not normal, they "flip the plane", taking us from "a right-hand universe" to a "left-hand universe". geometrically, this is the same reason the alternating subgroup is normal in S_n: it preserves parity.

many groups of small order have geometric interpretations as symmetry groups of objects one can actually look at, and it can be worth-while to actually do so. the cube and the tetrahedron are good places to start.